Class 8 Math Factorization notes
Factorisation – Class 8
Hi everyone! This chapter is all about Factorisation. It’s like the opposite of expanding algebraic expressions. Instead of multiplying things out, we break them down into simpler parts (factors).
What is Factorisation?
Factorisation is the process of writing an algebraic expression as a product of two or more expressions. Think of it like finding the prime factors of a number, but with algebraic expressions.
Example: x² + 5x + 6 can be factored as (x + 2)(x + 3).
Methods of Factorisation
Here are some common methods:
1. Common Factors:
Look for terms that have a common factor (number or variable) and factor it out.
Example: 2x + 4 = 2(x + 2) (2 is the common factor).
Example: xy + xz = x(y + z) (x is the common factor).
2. Grouping Terms:
Sometimes, you can group terms together to find common factors.
Example: xy + 2x + 3y + 6 = x(y + 2) + 3(y + 2) = (x + 3)(y + 2).
3. Using Identities:
Recognize expressions that fit known identities (like a² – b² = (a + b)(a – b), or a² + 2ab + b² = (a + b)²) and use them to factor.
Example: x² – 9 = (x + 3)(x – 3) (using the identity a² – b² = (a + b)(a – b)).
Example: x² + 6x + 9 = (x + 3)² (using the identity a² + 2ab + b² = (a + b)²).
4. Middle Term Splitting (for quadratic expressions):
For expressions like ax² + bx + c, find two numbers whose sum is ‘b’ and product is ‘ac’. Then split the middle term and factor by grouping.
Example: x² + 5x + 6. We need two numbers whose sum is 5 and product is 6. The numbers are 2 and 3.
x² + 5x + 6 = x² + 2x + 3x + 6 = x(x + 2) + 3(x + 2) = (x + 3)(x + 2).
Applications of Factorisation
1. Simplifying Algebraic Expressions:
Factorisation can make complex expressions easier to work with.
2. Solving Equations:
Factorisation is often used to solve quadratic equations and other polynomial equations.
3. Finding Areas and Volumes:
In geometry, factorisation can help find dimensions when the area or volume is given as an algebraic expression.
4. Problem Solving:
Many word problems in algebra can be solved more easily using factorisation.
Factorisation is a very important skill in algebra, so make sure you practice lots of examples!
Factorisation Quiz – Tough Application Problems
1. **Area of a Rectangle:** The area of a rectangular garden is given by the expression x² + 7x + 10. If the length of the garden is (x + 5), what is the width?
2. **Volume of a Cuboid:** The volume of a cuboid is given by the expression x³ + 6x² + 11x + 6. If the length is (x + 1) and the width is (x + 2), what is the height?
3. **Cost of Tiles:** A bathroom floor requires tiles. The total cost of the tiles is given by the expression 4x² + 12x + 9, where ‘x’ is related to the tile size. Factor the expression to determine possible dimensions related to the tiles.
4. **Simplifying a Rational Expression:** Simplify the expression: (x² – 4) / (x² + 4x + 4)
5. **Surface Area of a Cube:** The surface area of a cube is given by the expression 6x² + 24x + 24. What is the side length of the cube?
6. **Factoring a Cubic Expression:** Factor completely: x³ – 8
7. **Factoring a Trinomial:** Factor completely: 2x² + 5x – 3
8. **Area of a Triangle:** The area of a triangle is given by the expression (x³ + 5x² + 6x)/2. If the base of the triangle is (x + 2), what is the height?
9. **Simplifying a Complex Fraction:** Simplify: [(x² + 2x – 3)/(x + 3)] / [(x² – 1)/(x + 1)]
10. **Factoring by Grouping:** Factor completely: xy + 5x – 2y – 10